GeoProof — Nine-Point Circle explorer

side midpoints altitude feet orthocenter midpoints

9 points · 1 circle · radius = ½ × circumradius

Drag a vertex — even into an obtuse triangle (where the orthocenter and some feet move outside) — and all nine points stay on the circle.

The Nine-Point Circle

Every triangle has a special circle that passes through nine notable points all at once:

3 side midpoints + 3 altitude feet + 3 orthocenter–vertex midpoints

The "altitude feet" are where each altitude meets the opposite side; the last three are the midpoints of the segments joining the orthocenter H (where the altitudes meet) to each vertex.

Its size and place

The nine-point circle's radius is exactly half the circumradius R (the radius of the circle through A, B, C). Its center N is the midpoint of the segment from the circumcenter O to the orthocenter H — and it sits on the Euler line, along with O, H, and the centroid.

Why these nine?

The side midpoints and the orthocenter–vertex midpoints are related by a half-size scaling centered at H, which maps the circumcircle to the nine-point circle; the altitude feet land on it because each sees a relevant segment at a right angle. Three different-looking triples turn out to share one circle.

Try the obtuse case

Drag a vertex until a corner angle exceeds 90°. The orthocenter and two of the altitude feet leave the triangle — yet every one of the nine points remains exactly on the circle.

A bit of history

Often credited to Karl Wilhelm Feuerbach, who in 1822 proved the circle is also tangent to the triangle's incircle and excircles — so it is sometimes called the Feuerbach circle.